use of individual wheel steering to improve vehicle

Use of individual wheel steering to improve vehicle stability and disturbance rejection

by

Richard C.K. Nkhoma

Submitted in partial fulfilment of the requirements for the degree

MSc (Applied Science) Mechanical

in the Faculty of

Engineering, Built Environment and Information Technology

University of Pretoria, Pretoria

October 2009

©© UUnniivveerrssiittyy ooff PPrreettoorriiaa

Abstract

i

Use of individual wheel steering to improve vehicle stability

and disturbance rejection

By

Richard C.K. Nkhoma

Supervisor : Prof. N.J. Theron

Department : Mechanical and Aeronautical Engineering

Degree : MSc (Applied Science) Mechanical

ABSTRACT

The main aim of this research project is to extend theories of four-wheel-steering as

developed by J. Ackermann to include an individually steered four-wheel steering system

for passenger vehicles. Ackermann’s theories, including theories available in this subject

area, dwell much on vehicle system dynamics developed from what is called single track

model and some call it a bicycle model. In the bicycle model, the front two wheels are

bundled together. Similarly, the rear wheels are bundled together. The problem with this

is that it assumes two front wheels or two rear wheels to be under the same road, vehicle

and operating conditions. The reality on the ground and experiments that are conducted

are to the contrary. Therefore this study discusses vehicle disturbance rejection through

robust decoupling of yaw and lateral motions of the passenger vehicle.

A mathematical model was developed and simulated using Matlab R2008b. The model

was developed in such a way that conditions can be easily changed and simulated. The

Abstract

ii

model responded well to variations in road and vehicle conditions. Focus was in the

ability of the vehicle to reject external disturbances. To generate yaw moment during

braking, the brake on the left front wheel was disconnected. This was done because

lateral wind generators, as used by Ackermann, were not available. The results from both

simulations and experiments show disturbance rejection in the steady state.

Keywords – Disturbance rejection, yaw rate, lateral acceleration, four wheel steering

(4WS), individual wheel steering (IWS), robust control, robust decoupling.

Acknowledgement

iii

ACKNOWLEDGEMENT I would like to thank, Prof N. J. Theron, who not only served as my supervisor but also

encouraged and challenged me throughout my academic program and whose tireless

efforts and his ever present helping hand made this work a success. I would also like to

thank Prof. S. Els for his invaluable comments and inputs, and not forgetting my office

mate and now Dr. Michael Thoreson who tried all he could to supply solutions to all my

academic questions.

To my wife Catherine and child Peace for their encouragement and love despite the long

distance that separated us. You were a source of inspiration to me in times when the chips

were down.

To all friends too many to mention, I just want to say that thank you for all the moral

support that you rendered to me during my study, particularly I would like to thank

Jimmy Mokhafera for his support during the whole period of my research. May the Good

LORD richly bless you all.

Above all, I would like to thank God through Jesus Christ for HIS mercies and grace

towards me throughout my stay in South Africa.

Table of Contents

iv

TABLE OF CONTENTS

ABSTRACT ........................................................................................................................ i

ACKNOWLEDGEMENT ............................................................................................... iii

TABLE OF CONTENTS ................................................................................................ iv

LIST OF FIGURES ........................................................................................................ vii

LIST OF TABLES ........................................................................................................... ix

NOMENCLATURE ...........................................................................................................x

1 INTRODUCTION......................................................................................................1

1.1 Background ..........................................................................................................1

1.2 Statement of the Problem .....................................................................................2

1.3 Approach ..............................................................................................................2

2 LITERATURE REVIEW .........................................................................................4

2.1 Experimental Vehicle Status ................................................................................4

2.2 Current Research Situation ..................................................................................4

2.2.1 Decoupling controller ......................................................................................4

2.2.2 Drive by wire ...................................................................................................5

2.2.3 Driver assisted control .....................................................................................6

2.2.4 Adaptive Steering Controller ...........................................................................6

2.2.5 2H / H synthesis ..........................................................................................7

2.3 Closing .................................................................................................................9

3 MATHEMATICAL MODELLING .......................................................................10

3.1 Introduction ........................................................................................................10

3.2 Assumptions .......................................................................................................11

3.3 Nonlinear vehicle model equations of motion ...................................................12

3.4 Sideslip angles ...................................................................................................16

Table of Contents

v

3.5 Lateral forces .....................................................................................................18

3.6 Linearized model ...............................................................................................18

3.6.1 Linearized wheel sideslip angles ...................................................................18

3.6.2 Linearized cornering and lateral forces ..........................................................19

3.6.3 Linearized yaw moments ...............................................................................19

3.6.4 Linearized equation of motion .......................................................................20

3.7 Robust controller ................................................................................................22

3.8 The decoupled yaw subsystem...........................................................................26

3.9 Transfer functions ..............................................................................................28

3.9.1 Uncontrolled vehicle system ..........................................................................28

3.9.2 Controlled vehicle system ..............................................................................32

3.10 Position of centre of pressure .............................................................................36

3.11 Conclusion .........................................................................................................38

4 SIMULATIONS AND RESULTS ..........................................................................39

4.1 Decoupled car ....................................................................................................39

4.2 Conventional car ................................................................................................40

4.3 Results ................................................................................................................42

4.3.1 Validation of the model .................................................................................42

4.3.2 Simulations of a conventional and a decoupled vehicle ................................44

4.3.3 New control laws ...........................................................................................47

4.3.4 Simulations of different coefficient of friction .........................................50

5 IMPLEMENTATION OF INDIVIDUAL WHEEL STEERING .......................52

5.1 Introduction ........................................................................................................52

5.2 Motor characterisation .......................................................................................53

5.3 Potentiometer characterisation ...........................................................................57

5.4 DC motor controller ...........................................................................................61

Table of Contents

vi

5.5 Actuator assembly ..............................................................................................65

6 EXPERIMENTAL SETUP .....................................................................................68

6.1 Experimental Results .........................................................................................70

6.1.1 No input from the steering wheel...................................................................70

6.1.2 Sinusoidal input from the steering wheel .......................................................74

7 CONCLUSIONS AND FUTURE WORK .............................................................78

REFERENCES .................................................................................................................81

APPENDIX .......................................................................................................................85

List of Figures

vii

LIST OF FIGURES

Figure 1-1: SAE Vehicle axis system notations (SAE J670e (1976)) ........................... 3

Figure 3-1: Wheels turning the same direction ............................................................ 11

Figure 3-2: Sketch showing wheel sideslip angles ...................................................... 16

Figure 3-3: Decoupling point ....................................................................................... 22

Figure 3-4: Location of centre of pressure ................................................................... 37

Figure 4-1: Actuator model .......................................................................................... 41

Figure 4-2: simulation results at 50 1v m s ......................................................... 43

Figure 4-3: Conventional vehicle with front wheels steering only .............................. 44

Figure 4-4: Robustly decoupled vehicle with front wheel steering only ..................... 45

Figure 4-5: Front wheel steering and individual rear wheel steering .......................... 45

Figure 4-6: Force input response to different ......................................................... 47

Figure 5-1: Experimental vehicle ................................................................................ 53

Figure 5-2: Wiper motor sketch ................................................................................... 54

Figure 5-3: Motor tests setup ...................................................................................... 55

Figure 5-4: Motor response to a step input .................................................................. 55

Figure 5-5: Potentiometer connection sketch .............................................................. 58

Figure 5-6: Actuator unit ............................................................................................. 58

Figure 5-7: Graph of potentiometer angle against potentiometer volts ....................... 60

Figure 5-8: Graph of wheel angle against Potentiometer volts ................................... 61

Figure 5-9: Actuator block diagram. ............................................................................ 62

Figure 5-10: DC Motor Controller circuit diagram ....................................................... 63

Figure 5-11: DC motor Controller circuit box ............................................................... 63

Figure 5-12: Pulse Width Modulation, National Semiconductors (2005) ..................... 64

Figure 5-13: LMD18200 chip, National Semiconductors (2005) .................................. 64

Figure 5-14: LMD18200 circuit diagram, National Semiconductors (2005) ................ 65

Figure 5-15: Actuator – hub assembly connection ........................................................ 66

Figure 5-16: Actuator exploded view ............................................................................ 67

Figure 6-1: A sketch of experimental setup ................................................................. 68

Figure 6-2: Computer................................................................................................... 69

List of Figures

viii

Figure 6-3: Controlled vehicle yaw rate response. ...................................................... 71

Figure 6-4: Front right additional steering angle response .......................................... 72

Figure 6-5: Front left additional steering angle response ............................................ 72

Figure 6-6: Rear right steering angle response ............................................................ 73

Figure 6-7: Rear left steering angle response .............................................................. 73

Figure 6-8: Uncontrolled vehicle test result at 50 km/h .............................................. 75

Figure 6-9: Controlled vehicle test result at 50 km/h .................................................. 75

Figure 6-10: Steering angles response at low speed (20 km/h) ..................................... 76

Figure 6-11: Steering angles responses to braking at 40 km/h ...................................... 77

List of Tables

ix

LIST OF TABLES

Table B-1: Vehicle data ............................................................................................. 86

Nomenclature

x

NOMENCLATURE

i individual wheel slip angle

1 front left slip angle

2 front right slip angle

3 rear left slip angle

4 rear right slip angle

xa longitudinal acceleration

ya lateral acceleration

yDPa lateral acceleration at decoupling point

a distance the front axle is ahead of the centre of gravity (i.e., negative if the

centre of gravity lies ahead of the front axle)

A state dynamic matrix

fA frontal area of the vehicle

chassis sideslip

i chassis sideslip angle at individual wheel position

1 chassis sideslip angle at front left wheel

2 chassis sideslip angle at front right wheel

3 chassis sideslip angle at rear left wheel

4 chassis sideslip angle at rear right wheel

B state input matrix

Nomenclature

xi

iC individual wheel cornering stiffness

1C front left wheel cornering stiffness

2C front right wheel cornering stiffness

3C rear left wheel cornering stiffness

4C rear left wheel cornering stiffness

C state output matrix

dxC longitudinal drag coefficient

dyC lateral drag coefficient

MzC yaw moment coefficient

FyC lateral force coefficient.

i individual wheel steering angle

1 front left steering input (driver and control inputs)

2 front right steering input (driver and control inputs)

3 rear left steering input

4 rear right steering input

c control steering input

f steering wheel angle

s part of the front wheel steering angle that is the same for the left and right

wheels and is directly controlled by the driver

D state input output coupling matrix

Nomenclature

xii

dxF longitudinal disturbance force , assumed to be acting through the centre of

pressure.

yiF individual wheel force in the vehicle body axis system y direction

xiF individual wheel force in the vehicle body axis system x direction

dyF lateral disturbance force , assumed to be acting through the centre of

pressure

xF summation of all longitudinal forces

yF summation of all lateral forces

xtiF longitudinal wheel forces on the individual wheels axis system

ytiF lateral wheel forces on the individual wheels axis system

i corrective individual slip angle for front wheels only

cgh height to centre of gravity

iK generic gain parameter, where the index i is used to identify the gain in the

text.

zI moment of inertia about z axis

l wheelbase

fl distance the front axle is ahead of the centre of pressure (i.e., negative if

the centre of pressure lies ahead of the front axle)

Nomenclature

xiii

gpl distance the centre of pressure is ahead of the centre of gravity (i.e.,

negative if the centre of pressure lies behind the centre of gravity)

gml distance the centre of the vehicle is ahead of the centre of gravity

mpl distance the centre of pressure is ahead of the centre of the vehicle

rl distance the centre of gravity is ahead of the rear axle

m total mass

izM self aligning torques

ezM generic torques that are applied to the wheels

friction coefficient

r yaw rate and is the same as

refr reference yaw rate

defr the difference between reference yaw rate and the actual yaw rate

t half track width of the vehicle

t time

the angle of road inclination in the direction of travel.

sprocket angle. u state input vector

v vehicle velocity

xv longitudinal velocity

Nomenclature

xiv

yv lateral velocity

wrv resultant wind velocity

wyv crosswind velocity

0 0x y horizontal plane of an inertial axis system (as in figure 3-1)

x y horizontal plane of a vehicle body fixed axis system, x forward

(longitudinal), y lateral.

y state output vector

yaw angle

z vertical plane of the vehicle axis system, positive downwards

Chapter 1 Introduction

1

1 INTRODUCTION

1.1 Background

The concept of four wheel steering (4WS) in motor vehicles is manifested when the

driver is able to steer both front and rear wheels. There are several methods that have

been investigated in order to achieve 4WS, as noted in the literature review. There are

various reasons that necessitated the research into 4WS, like sudden disturbances

rejection (e.g. side wind forces, sudden wheel burst, rough roads, -split), improving

steerability and stability of vehicles and to increase ride comfort for the driver and

passengers.

Research of four-wheel car steering system has gained much attention from the early

1980’s. Since then, there has been a tremendously growing interest in the research and

development of 4WS. Early systems used simple open loop architecture to achieve active

control. To date several attempts have been made to improve handling characteristics and

performance of vehicles in order to increase manoeuvrability, stability, safety, and ride

comfort.

The use of single-track model to analyse the fundamentals of steering dates back as early

as 1940 as pointed out by Ackermann et al. (2002). This method assumes that steering

angles are the same for the two front wheels as well as for the two rear wheels. It is used

much in literature for the derivation of equations.

This research focuses on an extension of Ackermann’s theory, mainly as given by

Ackermann et al. (2002), to develop a system that would enable all wheels to be steered

individually. In addition to allowing each wheel to rotate through a steering angle, each


2

wheel is also equipped with a linear actuator to effect steering. The steering signal from

the driver is the conventional angular input using the steering wheel. The total steering

angle for a front wheel will thus be made up of the input from the driver and the angle

generated by the actuator. As for the rear wheels, the input angles will come from the

actuators only. The actuators are controlled by a control system, independently of the

driver. The purpose of this control system is to react to and reject suddenly applied

disturbances in the short delay period caused by the driver's slow reaction time, but then

to return control to the driver.

1.2 Statement of the Problem

This research analysed steering performance characteristics of a certain vehicle for all

speed ranges, i.e. at low speeds as well as at high speeds. Therefore this study

concentrated on theoretical and experimental analysis of the steering performance

characteristics of the vehicle under the above-mentioned conditions. This was done in

order to improve the current vehicle handling characteristics, manoeuvrability, stability,

safety, and to increase ride comfort for the driver as well as the passengers.

Measurements on the developed system indicated steady state rejection of disturbances.

Work of the previous researchers was investigated in order to assess the current level of

performance of 4WS as outlined in the subsequent chapters.

1.3 Approach

Theoretical modelling and analysis of individual wheel steering (IWS) was done when all

information was gathered both from available literature as well as from the existing 4WS

experimental vehicle in the SASOL Laboratory.


3

After developing the necessary theory, computer simulations were done and all

information gathered was applied to the experimental vehicle to observe the actual

performance. Therefore this research involved three sections:

Theoretical modelling and analysis of an IWS vehicle.

Computer simulations of an IWS vehicle.

Physical implementation and modifications on the existing four-wheel car

steering system, followed by road tests to evaluate the effects of modifications.

Figure 1-1 shows a standard SAE vehicle axis system and terminology as defined by SAE

J670e (1976). These are used throughout this work in the theoretical dynamic modelling.

The directions are as defined in the nomenclature.

Figure 1-1: SAE Vehicle axis system notations (SAE J670e (1976))

Chapter 2 Literature Review

4

2 LITERATURE REVIEW

2.1 Experimental Vehicle Status

The experimental 4WS vehicle that is available in the SASOL laboratory of the

University of Pretoria was designed and built by Burger in 1995. Klein (1996) used this

vehicle to study the optimisation of the phase shift in all wheel steering to minimize the

percentage overshoot on yaw velocity. He used active control system theory to achieve

active control of the vehicle’s steering system.

2.2 Current Research Situation

Different researchers have suggested various ways of achieving good characteristics and

handling performance of 4WS vehicles. Below are some of the suggested and

implemented methods, as found in the literature, from control point of view since the

focus of this research will mainly be on control issues.

2.2.1 Decoupling controller

Ackermann et al. (2002) (in particular chapter 6), make good observations and

analyses of the 4WS system. They firstly identify and then discuss decoupling

two steering tasks, for the purpose of improved steering control and disturbance

rejection. One of the tasks is to be performed by the driver, which is path tracking.

The other steering task is done by the automatic control system to counter the

effect of disturbance. This automatic control system is thought to do the

disturbance rejection faster and more precise than the driver. They achieved their

desired robust decoupling effect by cancellation of the yaw rate through a

feedback control law. This makes the yaw rate non-observable from the lateral

acceleration at the decoupling point. They showed through experiments that their


5

robust control system reduces the yaw rate to zero and then the driver, as his path

tracking task, returns the vehicle to the original heading.

The robust decoupling control concept is practically useful, only if the resulting

subsystems are stable, or can be stabilised separately without destroying the

decoupling effect. Some of the work about decoupling is also given in the papers

presented by Ackermann et al. (1992, 1995, 1996, 1997 and 2004).

Ackermann et al. (1999) say that there were some items that were not satisfactory

in the actual driving experiments through the use of robustly decoupling control.

The first one was that damping of the separated yaw dynamics was not sufficient

at high speeds. The second drawback was that integral feedback had been

implemented only to achieve robust unilateral decoupling despite providing

steady state accuracy. The last drawback outlined was that limit cycles happened

due to actuator rate limitations.

2.2.2 Drive by wire

Klein (1996) advocated the use of steer (drive) by wire. Lynch (2000) says that

with drive by wire, no mechanical restrictions exist in choosing the front and rear

steering angles and that the driver has no direct control over any of the wheels. In

this kind of steering, the steering wheel has no direct physical connections to the

wheels. All steering angles are supposed to be determined by an onboard

computer. One of the advantages of drive by wire is that there is tremendous

flexibility in designing the handling characteristics of the vehicle. Klein (1996)

applied in his research a strategy of yaw rate feedback to control rear steering


6

angles through the use of a controller, and driver input to control front steering

angles. Despite all this, drive by wire has its disadvantages and the obvious one is

the potential for disaster from controller failure as outlined by Lynch (2000).

2.2.3 Driver assisted control

Lynch (2000) proposes a concept of driver-assisted control (DAC) where the

driver has full command of the front wheel steering angle mechanically through

the steering wheel. He uses a flexible controller to improve the vehicle

performance by steering the rear wheels while allowing the driver to take full

charge in the event of controller failure. He incorporates the ideas behind drive by

wire and driver assisted control to achieve his goals.

The drawback is that his simulations focussed on low speeds in the range of 1 to

4 m s . His system was more oscillatory at higher speeds and this seems to

suggest that the system was moving towards instability at higher speeds.

2.2.4 Adaptive Steering Controller

Wu et al. (2001) used an adaptive controller for achieving accurate and prompt

control with noisy steering command signals and drifting valve characteristics on

an automated agricultural tractor with electro-hydraulic steering system. The

adaptive controller consisted of a feedforward base controller, a proportional-

integral-derivative (PID) base controller; a Kalman filter based adaptive (PID)

gain tuner, a wheel angle estimator and an adaptive nonlinearity compensator. In

this design, the feedforward controller determines the primary control signal on


7

the demand steering angle, and the PID controller provides a compensation signal

to offset the steering error based on the feedback signal.

The draw back in the adaptive control system is that there is a need to have a

process to identify real time vehicle response variables as pointed out by Abe

(1999, 2002). The problem then comes when the steering input from the driver is

very small, in which case the accuracy of the identification will deteriorate. On

top of that, the theoretical treatment of stability conditions for designing an

adaptive control system is complicated. To address these problems Wu focussed

on direct yaw moment control (DYC), where vehicle motion is controlled by a

yaw moment actively generated by the intentional excitation of wheel longitudinal

forces.

2.2.5 2H / H synthesis

Kitajima et al. (2000) used H control as an integral part of their design, which

optimises the control inputs and goals with predictable disturbances. In their

design, the front steering angle is considered as a detectable disturbance, whose

effect was to be rejected by the control signal. Their first integration design is a

feedforward integration type. In this design, one vehicle control input is

designated for each vehicle output and the other control inputs were treated as

disturbances. In order to study the effectiveness of these two integration designs, a

simulator, which realises vehicle longitudinal, lateral, roll, yaw and each wheel

rotational motion, was developed. In order to improve vehicle handling and

stability at high speeds, a multiobjective H optimal control was investigated by


8

Lv et al. (2004) based on yaw rate tracking. In particular, the four wheel steering

vehicle is controlled to simultaneously stabilise the responses of yaw rate, side

slip angle and lateral acceleration to the front wheel steering angle with the rear

wheels steered by wire.

You and Joeng (1998) designed an autopilot of a four-wheel steering vehicle

against external disturbances. To enhance the dynamic performance of this

automobile system, a mixed 2H / H synthesis with pole constraint was

designed on the basis of a full state feedback applying linear matrix inequality

(LMI) theory. For lateral/directional and roll motions, the steering angles were

actively controlled by steering wheel angles through the actuator dynamics.

Although the 2H approach is well suited to many real systems, it is known that its

stability and robustness cannot be guaranteed in the presence of various

uncertainties as pointed out by You and Joeng. As is the case with many vehicle

systems, a passenger car is expected to operate in a highly variable environment

and can be affected by fluctuations under manoeuvring conditions. This raises

questions about robustness of the control system by which the vehicle controller

must cope with these uncertainties successfully. They pointed out that H

synthesis guarantees a robust stability and disturbance rejection performance in

the presence of uncertainties. The drawback is that the H optimal controller

typically leads to an intolerable large control effort. To trade off, they combined

the two 2H and H effects to come up with 2H / H synthesis with pole

constraint via LMIs.


9

2.3 Closing

Some of the methods in the literature are tailored towards a particular variable. Various

researchers try to achieve better vehicle handling characteristics by designing controllers

for a specific situation like disturbance input from side wind forces or split, etc. So far

the solutions given in the literature are based on simplified and linearized models of the

vehicle systems. The drawback to this is that it is not possible to predict road conditions.

In this study, existing solutions have been analysed and incorporated to achieve better

vehicle handling performance characteristics and stability and to reject external

disturbances. Some of the variables neglected in most of the literature were considered.

This study wanted to specifically perform an investigation similar to what Ackerman, et

al. (2002) have done, but with a difference that instead of using a bicycle model, the

model used in this work has the two wheels on the same axle modelled separately, with

different conditions and the possibility for different steer angles and control signals. The

theory that was developed by Ackerman, et al. (2002), allows the control system to reject

disturbances caused by the two wheels on the same axle not experiencing the same

conditions, as illustrated by some of the experiments described in this book, like -split

braking. This theory works well for that case even though he does not model the left and

right wheels separately, and that illustrates the robustness of this theory. Furthermore,

this work will investigate whether modelling and controlling the left and right wheels

separately would not further improve the theory by Ackerman, et al. (2002), and to

investigate the benefits (or not) of having the possibility to have different control steering

angles at each of the four wheels. In the analysis, aerodynamic drag will be considered

and that the vehicle will be assumed to be travelling at constant speed.

Chapter 3 Mathematical modelling

10

3 MATHEMATICAL MODELLING

3.1 Introduction

To understand the behaviour of the vehicle at the point of turning, a model was developed

and using this model, equations of motion were derived. There are various methods of

achieving four-wheel vehicle steering as outlined by Lakkad (2004) and there are also

different ways of building models as well as ways of working on these models. In this

work, nonparallel steering was used for derivation of the equations of motion. This type

of steering is when the steered wheels on a single axle are not parallel during steering. It

has been chosen in order to achieve individual wheel steering where each wheel can be

steered towards the desired direction.

Figure 3-1 shows a scenario whereby the rear wheels turn in the same direction as the

front wheels. The direction of a positive steer angle is indicated for each of the wheels

in figure 3-1. The sketch shows the vehicle momentarily rotated at an angle with

respect to the 0 0x y axis system.

The vehicle will exhibit translational motion as well as rotational motion during a turning

manoeuvre. To describe the motion of the vehicle instantaneously, it is convenient to use

extra axes, fixed to and moving with the vehicle. With respect to the latter axes, the mass

moments of inertia of the vehicle are constant, where as with respect to the axes fixed in

space, the mass moment of inertia varies as the vehicle changes orientation, Wong

(1993). The x y z axis system is the vehicle coordinate system which is fixed with its

origin at the centre of gravity while 0 0x y z axis system is the non-rotating coordinate


11

system moving with the vehicle and this is shown at time t . The vehicle has rotated from

the 0 0x y axis system to the x y axis system by an angle , which is called yaw

angle.

Figure 3-1 by implication shows the definition of the different wheel axis systems, which

for wheel i is rotated about the z-axis through the steering angle i , with respect to the

vehicle coordinate system.

2

r v

xv

yv

1

4

3

FR

ON

T

RE

AR

dyFy0y

0x

x

3ytF3xtF

4ytF4xtF

2xtF

1xtF1ytF

2ytF

dxFyDPa

Figure 3-1: Wheels turning the same direction

3.2 Assumptions

In order to simplify the derivation of the equations of motions, the following assumptions

are made:

i. All the analyses that will be done will assume that the vehicle is travelling or

being driven at a constant speed.

ii. The force in the x direction of the wheel axis system is sufficient to balance

drag but is assumed to be significantly smaller than the lateral direction

forces.


12

iii. The vehicle is turning at constant velocity and with this assumption

acceleration in the longitudinal direction is negligible.

iv. The vehicle model to be considered is a two dimensional and rigid body

vehicle.

v. The normal force of the vehicle is distributed equally onto the left and right

wheels and the forces transferred by the wheels are applied in the centre of the

wheel contact patch.

vi. Effects from the suspension and wheel deformation were neglected.

vii. The vehicle is travelling on a flat surface i.e. x y plane and motions to be

considered occur along this plane.

viii. The vehicle is symmetrical about x z plane and the centre of pressure and

the centre of gravity is in this plane.

ix. Roll, pitch and translational motion along the z axis was neglected and the

remaining three degrees of freedom were considered i.e. longitudinal motion

along the x axis , lateral motion along the y axis and yaw motion around

the z axis .

x. Track width 2t at the rear is the same as at the front.

3.3 Nonlinear vehicle model equations of motion

In this section, equations of motion that describe individual wheel steering are derived.

Referring to figure 3-1:

Longitudinal forces for the front and rear wheels are:

cos sinxi xti i yti iF F F [3.1]


13

Lateral forces for the front and rear wheels are:

sin cos yi xti i yti iF F F [3.2]

where 1, 2,3,4i

The longitudinal and lateral vehicle velocities can be calculated from the actual velocity

v which is at a chassis sideslip angle to the vehicle’s longitudinal axis. The

corresponding acceleration components can be found by differentiating these velocities,

i.e.

cosxv v cos sinxv v v [3.3]

sinyv v sin cosyv v v [3.4]

Let f gpa l l [3.5]

Newton – Euler’s laws can now be applied to figure 3-1 in order to derive equations of

motion. More details about some of the derivations can be found in the books by Wong

(1993), Gillespie (1992) and Genta (1997).

Translational motion

The sum of the external forces acting on the body in a given direction is equal to the

product of its mass and the acceleration of the C.G. in that direction, i.e.

o longitudinal motion in the directionx xF ma x [3.6]

o lateral motion in the directiony yF ma y [3.7]

The summation of acceleration should take into account the effect of yaw rate to give a

complete picture of acceleration in the x and y direction.


14

a. Longitudinal direction

xx xi d xF F F ma [3.8]

Where 21

2dx f wr dxF A v C is typically the drag force. If the vehicle is travelling at an

inclined plane, then the effect of weight in the form of sinmg is added (Genta, 1997).

x x ya v rv [3.9]

xx y xi dm v rv F F [3.10]

Substituting for xv and yv from equations [3.3] and [3.4], equation [3.8] now becomes:

cos sinxxi dmv mv r F F [3.11]

b. Lateral direction

yy yi d yF F F ma [3.12]

y y xa v rv [3.13]

yy x yi dm v rv F F [3.14]

Substituting for yv and xv , equation [3.12] now becomes:

sin cosyyi dmv mv r F F where 1,2,3,4i [3.15]

c. Rotational motion

Gillespie (1992) points out that the sum of the torques acting on a body in a given

direction is equal to the product of its rotational moment of inertia about an axis through

its C.G., and the rotational acceleration about that axis.

z zM I r [3.16]


15

1 2 3 4 2 4 1 3z y y r y y x x x x dy gp D zM a F F l F F t F F F F F l M I r

[3.17]

where according to Genta (1997)

i. As said before, ydF is the component of aerodynamic forces in the y direction. The

resulting wind velocity when driving, wrv , is the combination of the apparent wind

velocity due to the vehicle’s forward motion, xv , and the cross wind velocity, wyv ,

which is written as: 2 2wr x wyv v v .

The resulting airflow has an angle of approach with respect to the vehicle, , which

is computed as:

arctan wy

x

v

v

Hence:

21

2y yd wr f dF v A C , (Hucho 1987, p.63).

ii. DM is made up of all the moments that are developed due to disturbance inputs i.e.

21

2i z eD z wr f M zi

M M l v A C M (Hucho 1987, p, 63).

Using equations [3.11], [3.15] and [3.17] to solve for , r and v gives:

sin cos 0

cos sin 0

0 0 1

xi dx

yi dy

z

mv r F F

mv F F

I r Y

Where

1 2 3 4 2 4 1 3y y r y y x x x x dy gp DY a F F l F F t F F F F F l M


16

But

1sin cos 0 sin cos 0

cos sin 0 cos sin 0

0 0 1 0 0 1

therefore

sin cos 0

cos sin 0

0 0 1

xi dx

yi dy

z

mv r F F

mv F F

I r Y

[3.18]

3.4 Sideslip angles

Wheel slip angle is defined as the angle that is so formed between the x – axis of the

wheel axis system and the actual direction of the wheel velocity, while the sideslip angle

is the angle between the vehicle’s actual velocity vector v and the vehicle axis x .

Figure 3-2 shows individual wheel sideslip angles as well as the front and rear chassis

sideslip angles.

r

v

xv

flvyv

1

FR

ON

T

RE

AR

ydF

1

1 1c

s

frv

22

2 2c

s2f

1f

rrv 44

4

rlv 33

3

Horizontal line in the

plane of the wheel


plane of the wheel


plane of the wheel


plane of the wheelxd

F

Figure 3-2: Sketch showing wheel sideslip angles


17

From figure 3-2, it can be noted that:

i i i where 1, 2,3, 4i [3.19]

and that

i s i where 1, 2i for front wheels.

For the calculation of sideslip angles, the equations listed below were used as reported by

Ghelardoni (2004) as well as Zhengqi et al. (2003) and are modified according to this

work. Referring to figure 3-2:

1 1 1tan tany

x

v ar

v tr

[3.20]

2 2 2tan tany

x

v ar

v tr

[3.21]

3 3 3tan tany r

x

v l r

v tr

[3.22]

4 4 4tan tany r

x

v l r

v tr

[3.23]

From where the individual wheels sideslip angles can be found to be:

11 1

sintan

cos

v ar

v tr

[3.24]

12 2

sintan

cos

v ar

v tr

[3.25]

13 3

sintan

cosrv l r

v tr

[3.26]

14 4

sintan

cosrv l r

v tr

[3.27]


18

3.5 Lateral forces

Lateral forces that are developed at the contact patch between the individual wheels and

the ground are normally referred to as cornering forces. According to Ackermann, et al.

(2002, equation (6.4.1)), the cornering forces are functions of the sideslip angles. They

state that the relationship between the lateral force and the wheel slip angle, when the

camber angle of the wheel is zero and when no sliding is taking place, is given as:

cosyi i i i iF C where 1, 2,3, 4i [3.28]

3.6 Linearized model

Most equations derived so far are nonlinear due to the presence of trigonometrical

functions. Nonlinearity is also coming from the fact that naturally wheel forces are not

linear. Ackermann et al. (2002) say that in normal driving situations (except slow parking

manoeuvres), the most important nonlinearity is the uncertain wheel model. If small

angles are assumed, like if we assume small steering angles, small sideslip angles of the

wheels and small sideslip angle of the vehicle, we know that tan sin and

cos 1 where stands for a small angle.

3.6.1 Linearized wheel sideslip angles

According to You et al. (1998), if small angles are assumed, xv t r . Therefore

x xv tr v and x xv tr v . Also xv v

From equation [3.24] to [3.27], the linearized wheel sideslip angles’ equations are:

i i i ix x

v ar v ar ar

v tr v v

where 1, 2i [3.29]

r r ri i i i

x x

v l r v l r l r

v tr v v

where 3, 4i [3.30]


19

3.6.2 Linearized cornering and lateral forces

The second equation in equation [3.18] deals with the longitudinal acceleration of the

vehicle. The analysis may be limited to the case where the vehicle is driven at a constant

speed, which firstly means some propulsive force is necessary in the x direction to

balance the effect of the aerodynamic drag force, and secondly that this equation need not

be further considered. Turning to the first equation in equation [3.18], the resultant x -

direction force xi dxF F , if not zero, would be very small and the multiplication of this

small quantity with the sine of the small sideslip angle can be ignored. Therefore, the

linearized form of the first equation in equation [3.18] reduces to:

1 2 3 4 yy y y y dmv r F F F F F [3.31]

From equation [3.28], it follows that the linearized lateral forces in the vehicle axis

system, based on the assumption that i is small, are:

yi i i i i i i iar

F C C Cv

where 1, 2i [3.32]

ryi i i i i i i i

l rF C C C

v where 3, 4i [3.33]

3.6.3 Linearized yaw moments

The vehicle x -direction forces can be ignored based on the argument that they are quite

small, and are also multiplied with a moment arm t which is typically half the length of

the moment arms a and rl of the lateral forces. Then the yaw motion equation reduces

to:

1 2 3 4z y y r y r y dy gp DI r aF aF l F l F F l M [3.34]


20

3.6.4 Linearized equation of motion

The final linearized equations of motion will now be equations [3.31] and [3.34], which

can now be summarised in matrix form as:

1 2 3 4

1 2 3 4

yy y y y d

y y r y r y dy gp Dz

F F F F Fmv r

aF aF l F l F F l MI r

[3.35]

Using equations [3.32] and [3.33] of the relationships between lateral forces and wheel

slip angles, we have:

1 1 1 1 1 2 2 2 2 2

3 3 3 3 3 4 4 4 4 4

1[

]y

r rd

ar arC C C C

mv v v

l r l rC C C C F r

v v

1 1 1 1 1 2 2 2 2 2

3 3 3 3 3 4 4 4 4 4 g

1[

]

z

r rr r r r dy p D

ar arr a C a C a C a C

I v v

l r l rl C l C l C l C F l M

v v

[3.36]

In matrix form:

11 1 2 2 3 3 4 4

1 2 2

gp1 1 2 2 3 3 4 43 4 3

4

10

C

1Cyd

r r D

z z z z z z

C C C CFC mvmv mv mv mv

la C a C l C l CC rr MI I I I I I

[3.37]

Where:

1 1 1 2 2 3 3 4 41

C C C C Cmv [3.38]


21

2 1 1 2 2 3 3 4 42

11r rC a C a C l C l C

mv [3.39]

3 1 1 2 2 3 3 4 41

r rz

C a C a C l C l CI [3.40]

2 2 2 24 1 1 2 2 3 3 4 4

1r r

z

C a C a C l C l CI v [3.41]

i.e. this is in the form of a state space equation and using Friedland (1986) notations,

equation [3.37] can be written as:

dx x u x A B E and [3.42]

y x u C D [3.43]

Where

1 2

3 4

C

C

C

C

A

1 1 2 2 3 3 4 4

1 1 2 2 3 3 4 4r r

z z z z

C C C C

mv mv mv mva C a C l C l C

I I I I

B

gp

1 0

1

z z

mvl

I I

E xr

1

2

3

4

u

ydd

D

Fx

M

[3.44]

The output from this linearized model, given by equation [3.43], is the vector of lateral

acceleration at the decoupling point and yaw rate as shown later.


22

3.7 Robust controller

Ackermann et al. (2002 pp. 177,178) suggested the implementation of a robust controller

to aid the driver in his steering task and this section is essentially based on their

suggestions. To begin with, they explain that there exist a point called a decoupling

point, as shown in Figure 3-3, which experiences a lateral acceleration yDPa . This

position is used to decouple two steering tasks of the vehicle, which are path tracking by

the driver and the automatically controlled yaw stabilization and disturbance

compensation. Ackermann et al. (2002), further explain that the indirect influence of the

disturbance torques on the lateral acceleration yDPa via the vehicle dynamics should be

compensated such that the driver controls the undisturbed yDPa in his or her path

tracking task. They pointed out that in system theoretical terms, the task separation

requires the yaw rate r to be non-observable from yDPa . The condition for this

decoupling is that DP z rl I ml . This may be shown as follows, with reference to figure

3-3:

FR

ON

T

RE

AR

dyF

Figure 3-3: Decoupling point


23

yDP yCG DPr

a a lt

1 2 3 4yCG y y y y dyma F F F F F

1 2 3 4z y y f gp y y r dy gp Dr

I F F l l F F l F l Mt

1 2 1 2 3 4 3 4y y y y f gp DP y y y y r DPyDP

z z

dy gp DP dyDPD

z z

F F F F l l l F F F F l la

m I m I

F l l FlM

I I m

[3.45]

By defining the decoupling point such that

DP z rl I ml [3.46]

the contribution of the lateral forces at the rear wheels to the acceleration of the

decoupling point is zero, so that

1 2 1 2 g.y y y y f gp DP dy p DP dyDPyDP D

z z z

F F F F l l l F l l Fla M

m I I I m

g1 2

1 f gp dy p dyDy y

r r r

l l F l FMF F

m ml ml ml m

g1 2

11 f gp dy p dyD

y yr r r

l l F l FMF F

m l ml ml m

g1 2

1 dy p dyDyDP y y

r r r

F l FMa F F l

ml ml ml m [3.47]

Hence

2

1yDP yi D dy r gp r

i

a F l M F l l ml

[3.48]


24

2

1i i i D dy r gp r

i

C ar v l M F l l ml

[3.49]

Ackermann at al. (2002), stress that this unilateral decoupling must be robust for all

operating conditions. The feedback control law used to make the yaw rate r non-

observable from the lateral acceleration yDPa at the decoupling point, as suggested by

these authors (Ackermann at al. 2002, p. 186), is:

c DPref

l ar r r

t v

[3.50]

Making zI in equation [3.46] subject of formula and substituting this relationship into

equation [3.35], we have:

1 2 3 4

1 2 3 4

yy y y y d

y y r y r y dy gp Dr DP

mv r F F F F Ft

aF aF l F l F F l Mrml l

t

[3.51]

Solving equation [3.51] by first multiplying the top equation by rl and then adding the

two equations, followed by multiplying the top equation by a and then subtracting the

bottom equation from the top equation, results in the following two simultaneous

equations:

1 2r DP y y dy r dy gp Dr

ml v r l F l F l F l F l Mt t

[3.52]

3 4r DP y y dy dy gp Dr

m va r l l F l F l F a F l Mt t

[3.53]

Substituting for r from equation [3.50]

cref DP

rr r l a v

tt

[3.54]


25

into equation [3.52], the result is

1 2c

r y y dy r dy gp D r refr

mvl a v F l F l F l F l M mvl rtt t

[3.55]

The linearization that renders equations [3.29] and [3.30] from equations [3.24] to [3.27]

also implies that, after linearization, 1 2 and 3 4 . Also, because the control law

enforces 1 2c c c , there are no longer two distinctive angles after linearization

i.e., 1 2f f . Therefore, from now onwards, whenever the linearized model is

discussed or used f will be used for both 1 and 2 , r will be used for both 3 and

4 , and f for both 1f and2f

.

From f f c and substituting f ar v from front wheels equation(see

equation [3.29]), we have

f car v , which will give f equal to the expression in the square brackets of

equation [3.55]. Substituting f for this expression leads to:

1 2f

y y dy r gp D r refF l F l F l l M mvl rt

[3.56]

Also from equation [3.48], it can be seen that:

fyDP refa v r

t

[3.57]

It can be noted from equation [3.56] that this first order equation in does not depend on

the state variable r . This shows that the control law, equation [3.50], makes the yaw rate

non-observable from the lateral acceleration yDPa at the decoupling point. The lateral


26

forces are functions of slip angles i.e. 1 1 1y yF F and 2 2 2y yF F . From figure 3-2,

1 s f and thus independent of r . Therefore 1y s fF = 2y s fF . This

leads equation [3.56] to:

1 2f

y s f y s f dy r gp D r refF l F l F l l M mvl rt

[3.58] 3.8 The decoupled yaw subsystem

From the linearization that produces equation [3.30] from equations [3.26] and [3.27]

(see also the introduction of r just after equation [3.55]), we can see that:

/r rl r v [3.59]

Therefore

/r rl r v

Substituting for from equation [3.53], we have:

3 4

1 r DPr y y dy gp D

l l alF lF F a l M r r

mva va

[3.60]

Solving for r

t

from second row of equation [3.51], we have:

1 2 3 41

y y r y r y dy gp Dr DP

raF aF l F l F F l M

t ml l

[3.61]

From equation [3.48], we have

2

1

yi yDP r D dy r gp

i

F a ml M F l l l

[3.62]

When equation [3.62] is substituted into equation [3.61], we have:


27

3 41

yDP r D dy r gp r y r y dy gp Dr DP

ra a ml M F l l l l F l F F l M

t ml l

3 4dy r gpyDP y y dy gpD D

DP r DP r DP DP r DP r DP

aF l laa F F F laM Mr

t ll ml l l ml l l ml ml l ml l

3 41

r gpyDP y y dy Dgp

DP DP r DP r DP

a l laa F F F Mr al

t ll ml ml l l ml l l

3 4yDP y y dy f D

DP DP DP DP

aa F F F l Mr

t ll ml mll mll

[3.63]

where r gp fl l l l and rl a l

Substituting equation [3.63] into equation [3.60] will yield the following yaw subsystem

equation in state space form.

4

3

/1

1DP rr

yi DDP i

l l vF M l

mlr

1

1 /

0

dyDP DP gp r DP f r f

DPf

yDPr DP

DP

Fll a ll l l l l l l ava

mlll

al l a vr

lla

4

3

/1

1DP rr

yi DDP i

l l vF M l

mlr

1 1 /

0

Fdy yDPr DP

DP DPf

F al l a vrva mll lla

l

[3.64]

Where 1

DP DP gp r DP f r fFll a ll l l l l l l a


28

Equation [3.64] is similar to what Ackermann et. al (2002) found with few additionals in

the terms of aerodynamics forces dyF and distance terms. The quantity yDPa is used as

the coupling term as it may be measured with accelerometers. yDPa is also used in the

task separation.

3.9 Transfer functions

This section discusses the transfer functions of the two systems which are the vehicle that

does not have any controlling measures and the vehicle with controlling measures. We

will begin our discussions with an uncontrolled vehicle system and later on we will talk

about a controlled vehicle system.

3.9.1 Uncontrolled vehicle system

From equation [3.37] the characteristic polynomial would be:

1 4 2 3P s s C s C C C [3.65]

21 2s c s c where [3.66]

1 4 1c C C

2 1 4 2 3c C C C C

Also, the resolvent is given by:

1 4 2

3 1

1 s C Cs

C s CP s

I A [3.67]

The uncontrolled (undecoupled) transfer functions will now be derived.

Equation [3.37] may also be written as:


29

1

21 1 2 2 3 3 4 4

31 2

gp 41 1 2 2 3 3 4 43 4

10

C

1C

y

r r

dz z z z z z

D

C C C C

C mv mv mv mv mvla C a C l C l CC rr

FI I I I I I

M

[3.68]

Which is in the standard form

x Ax Bu [3.69]

Where the state vector x and matrix A are defined as in equation [3.44], and

1 1 2 2 3 3 4 4

gp1 1 2 2 3 3 4 4

10

1r r

z z z z z z

C C C C

mv mv mv mv mvB

la C a C l C l C

I I I I I I

and

1 2 3 4

T

dy Du F M

Let the output be r and yDPa or

yDP

ry

a

[3.70]

Then the output equation is in the form:

y Cx Du

Using equation [3.49]

2 2

1 1

0 1

/i i i i

i ir r

C l al vC C

ml ml

and


30

1 1 2 2

0 0 0 0 0 0

10 0 r gp

r r r r

l lD lC lC

ml ml ml ml

The Laplace transformation of equation [3.69] is:

1

x s s Ax s Bu

sI A x s Bu

x s sI A Bu

[3.71]

Substituting equation [3.71] into the Laplace transformation of equation [3.70], we have:

1

1

y s Cx s Du

C sI A Bu Du

C sI A B D u

[3.72]

This can be expressed in the following form:

1

2

31 2 3 4 5 6 11 1 2 2

47 8 9 10 11 12 2

0 01

y

yDPr r

d

D

s

s

sr s b b b b b b slC lCsa s b b b b b b sP s

ml mlF s

M s

0 0

1dy

gpD

r r

F slr l

M sml ml

[3.73]

Where:

1 1 13 1 11

z

s C a CC Cb

mv I

; 1 2 23 2 2

2z

s C a CC Cb

mv I

;

1 3 33 3 33

r

z

s C l CC Cb

mv I

; 1 4 43 4 4

4r

z

s C l CC Cb

mv I

;


31

35 1

gp

z

lCb s C

mv I ;

16

z

s Cb

I

; 7 1 1

jk

z

ajkb C

mv I

;

8 2 2jk

z

ajkb C

mv I

; 9 3 3r jk

z

l jkb C

mv I

;

10 4 4r jk

z

l jkb C

mv I

; 11gp jk

z

l jkb

mv I

; 12j

z

jb

I ;

4 3

/i i i ik

r r

l C a v Ck s C C

ml ml

and

2 1

/i i i ij

r r

l C a v Cj C s C

ml ml

where for both kk and jj , the summation is

for i from 1 to 2.

This can be reduced to:

1

2

31 2 3 4 5 6

47 8 9 10 11 12

y

yDP

d

D

s

s

sr s d d d d d dP s sa s d d d d d d

F s

M s

[3.74]

Where 1d to 6d will be the same as 1b to 6b respectively and:

1 17 7

r

P s Cd b

ml

; 2 2

8 8r

P s Cd b

ml

;

11 11r gp

r

P s l ld b

ml

and

12 12

r

P sd b

ml while 9d and 10d will remain as 9b and 10b respectively.


32

3.9.2 Controlled vehicle system

From equation [3.58] we can substitute the linearized equivalent of

yi s f i i s fF C and the results are as shown below:

1 2f

y y dy r gp D r refF l F l F l l M mvl rt

2

1

fi i i dy r gp D r ref

i

l C F l l M mvl rt

1 1 2 2f

s f dy r gp D r refC C l F l l M mvl rt

[3.75]

From equation [3.57], we have

fyDP refa v r

t

1 1 2 2 1 1 2 2 dy r gpf s D

r r r r

F l ll C C l C C M

ml ml ml ml

[3.76]

Substituting equation [3.76] into equation [3.64] with the help of equations [3.33] and

[3.59], we have:

5 6 7 81 3

9 10 11 122 4

1

0

s

rrrf

dy

D

a a a aa a

Fa a a aa arr

M

[3.77]

where for the time being, 3 4 r . Later on, in section 4.2.3.1, the concept of

independent steering angles on both front and rear wheels is re-introduced.


33

4

13

DP ri i

iDP

l la C

mvl

;

4

23

1i i

iDP

a Cml

; 2

31

DPi i

iDP

l aa C

mvl

2

41

i iir DP

aa C

ml l

; 2

51

DPi i

iDP

l aa C

mvl

;

4

63

DP ri i

iDP

l la C

mvl

;

2 17

F F

DP

aa

mvall

; 8

1

DP

amvl

; 2

91

i iir DP

aa C

ml l

;

4

103

1i i

iDP

a Cml

; 11gp

DP r

la

ml l 12

1

r DP

aml l

;

Where 2F DP r gpl a l l

The above state space equation of the robustly decoupled vehicle, equation [3.77],

together with equation [3.75], can be grouped into:

1 3 5 6 7 8

2 4 9 10 11 12

13 14 15 16

1 0

0 0

0 0 0 1

s

rrr

dy

ff D

ref

a a a a a aFrr a a a a a a

a a a a M

r

[3.78]

Where:

2

131

i iir

la C

mvl

; 2

141

i iir

la C

mvl

;

15r gp

r

l la

mvl

; 16

1

r

amvl

;

The decoupled output equation is:


34

2 3 41

0 0 0 0 00 1 0

0 00 0

s

rr

dyyDP

f D

ref

rFr

a f f ffM

r

[3.79]

where:

2

11

i iir

lf C

ml

; 2

21

i iir

lf C

ml

;

3r gp

r

l lf

ml

; 4

1

r

fml

;

Equations [3.78] and [3.79] are recognised as a 3rd order state space model of the

decoupled controlled system, of a form similar to equations [3.68] and [3.70], with state

vector T

r fr , input vector T

s r dy D refF M r i.e., the vector on the far

right of equation [3.79]), the output vector T

yDPr a and the matrices given by:

1 3

2 4

13

1

0

0 0

a a

A a a

a

5 6 7 8

9 10 11 12

14 15 16

0

0

0 1

a a a a

B a a a a

a a a

1

0 1 0

0 0C

f

2 3 4

0 0 0 0 0

0 0D

f f f

Transforming equations [3.78] and [3.79] to the Laplace domain leads to the transfer

functions 1g to 9g , defined by:


35

1 2 3 4 5

6 7 8 90

s

r

dyyDP

D

ref

r g g g g gF

a g g g gM

r

[3.80]

Where:

1 1 2 21 2

1 1 2 2r DP

s C C v amvs qg

mvl s lC lC mvl s ps z

10 2 6 10 12 2

1 2

a s a a a ag

s a s a

211 15 4 2 7 11 13 11 1 15 3 2 15 4 1 2 7 13 11 1 13

3 3 213 1 1 13 2 2 13

a s a a a a a a a a s a a a a a a a a a a a ag

s a a s a a a s a a

212 16 4 2 8 12 13 12 1 16 3 2 16 4 1 2 8 13 12 1 13

4 3 213 1 1 13 2 2 13

a s a a a a a a a a s a a a a a a a a a a a ag

s a a s a a a s a a

1 1 2 25 2

1 1 2 2r DP

C C v amvs qg

mvl s lC lC mvl s ps z

1 1 2 26

1 1 2 2r

vC vC sg

smvl lC lC

21 1 1 1 2 2 2 2 2 1 2 2

71 1 2 2

r gp r gp r r r

r r

C l C l C l C l smvl l lC l lCg

smvl lC lC ml

1 1 2 2 1 1 2 2

81 1 2 2

r

r r

C C smvl lC lCg

smvl lC lC ml

1 1 2 29

1 1 2 2r

C C vg

smvl lC lC

and


36

3 3 4 4 3 3 4 4Dp Dp r rp l C l C l C l C ,

3 3 4 4 3 3 4 4r rq a C a C l C l C ,

3 3 4 4z C C v

Some of the g-functions are expressed in terms of the basic parameters to highlight

further simplification, whereas others are expressed in terms of the a-coefficients of

equations [3.77] and [3.78] for the sake of simplicity without which they looked clumsy.

Equation [3.80] can be rewritten as:

1 2 3 4

6 7 80

s

r

yDP dy

D

ref

r g g g g

a g g g F

M

rs

[3.81]

3.10 Position of centre of pressure

Hanke et al. (2001) conducted research on analysis and control of vehicle dynamics under

cross wind conditions. As already pointed out, the resultant crosswind effect is thought to

act at the centre of pressure. Generally, determining the centre of pressure can be a very

complicated procedure since the pressure distribution is bound to change around the

object under varying conditions. The position of the centre of pressure also depends on

the type of the vehicle body. Hanke et al. (2001) show that, usually, the position of centre

of pressure lies in the front half of the vehicle between the front axle and centre of

gravity. This is in line with the explanation given by Hucho (1987), which states that

there exists a place M which is called the aerodynamic reference point. This point is

located in the middle of the wheel base and the middle of the track. Since the location of


37

C.G. of the experimental vehicle is way beyond the centre of the vehicle, M was assumed

to lie between C.G. and C.P. Eventually the equation for deriving the distance between

C.P. and C.G. will be the same as given by Hanke (2001). F

RO

NT

RE

AR

Figure 3-4: Location of centre of pressure

Mzmp

Fy

Cl l

C [3.82]

Then the equation between C.P. and C.G. can be found algebraically from:

2gp mpl

l a l , this gives

2Mz

gpFy

Cll a l

C [3.83]

where these variables are as defined under nomenclature.

From the tables given in Gillespie (1992), the ratio of yaw moment coefficient to lateral

force coefficient can be calculated to be approximately equal to 0.2 .

i.e.


38

0.2Mz

Fy

C

C [3.84]

3.11 Conclusion

This chapter was all about mathematical modelling of the experimental vehicle which

will be used in the simulations and experimental tests.

Chapter 4 Simulations and Results

39

4 SIMULATIONS AND RESULTS

This section deals with the relationships between the input and output of the decoupled

car system and conventional car system according to the equations derived in chapter 3.

From the relationships that are derived, the equations are simulated and results are

presented.

4.1 Decoupled car

The characteristic polynomial of the decoupled car is derived from equation [3.78] and

can be written as:

213 1 2s a s a s a with meanings according to Ackermann et al. (2002, pp.207-208)

i.e.

13s a is the lateral characteristic polynomial and

21 2s a s a is the yaw characteristic polynomial.

The yaw characteristic polynomial will be written as:

2

2

1 2

4

43

3

yaw

i iDP r i

i iiDP DP

P s a s a

Cl l

s C smvl ml

[4.1]

In conventional second order way of writing equations, equation [4.1] can be written as:

2 22yaw n nP s s [4.2]

with

4

3i i

in

DP

C

ml

and [4.3]


40

4

3

2

i iDP r i

DP

Cl l

v ml

[4.4]

Equation [4.4] produces a small damping, this makes the yaw dynamics to oscillate. To

have the meaningful results, the damping coefficient in the yaw dynamics equation has to

be changed. According to Ackermann et al. (2004), the yaw dynamics can be stabilised

by making large.

For the decoupled car, the transfer functions will be, (see equation [3.81]):

1

s

r sG s g

s [4.5]

3F

dy

r sG s g

F s [4.6]

4M

D

r sG s g

M s [4.7]

4.2 Conventional car

From equation [3.74], the transfer function of the conventional car from 1 and 2 to r is:

11

1

3 1 1 1 1 12

1 2

z

r s dG s

s P s

C C mv s C a C I

s c s c

3 1 1 1 1 1 1 1

21 2

z zC C mv C a C I a C I s

s c s c

[4.8]

similarly


41

22

2

3 2 2 1 2 22

1 2

z

r s dG s

s P s

C C mv s C a C I

s c s c

3 2 2 1 2 2 2 2

21 2

z zC C mv C a C I a C I s

s c s c

[4.9]

Where P s is as explained in equation [3.65]

From Ackermann et al. (2002, pg 234), it is stated when discussing comparative

simulations, that for the sake of comparability, the conventional car is assumed to be a

steer-by-wire vehicle equipped with the same steering actuator as the controlled car.

Therefore, the transfer function from steering wheel angle to yaw rate r could be found

by multiplying the transfer function as in equations [4.8] and [4.9] with the transfer

function of the actuator dynamics i.e. from equation [3.74]:

1 2 a

f

r sG s G G G

s

Where aG is for actuator dynamics. The transfer function can be seen as follows:

2K1

s

+_f 1K

1

K

s

Motor

3K

iSprocket

angle

Figure 4-1: Actuator model

21 3

Ba

KG

s s K K K

with


42

1 2BK K K K

The transfer functions for yaw disturbance input from lateral disturbance force dyF and

yaw disturbance moment, DM , will be, (see equation [3.74]):

5

3 1

21 2

Fdy

gp z

r s dG s

F s P s

C mv s C l I

s c s c

[4.10]

3 1

21 2

gp z gp zC mv C l I l I s

s c s c

and

6

12

1 2

MD

z

r s dG s

M s P s

s C I

s c s c

12

1 2

z zs I C I

s c s c

[4.11]

4.3 Results

This section deals with simulated results that were done. The vehicle step responses are

shown where the inputs were lateral force input and yaw disturbance torque.

4.3.1 Validation of the model

In order to check the validity of the model and the code, the system was given similar

inputs and parameters to what Ackermann et al. (2002) did and the results are as shown

in figure 4-2. The simulated results as seen in the graphs compare very well with what is

found in the Ackermann et al. (2002), especially on page 235. The value for torque


43

disturbance input used was 1300Nm while the steering wheel angle step input was 0.13 .

There is a good correlation between these two results and this stems from the fact that the

two models were analysing the same physical system. Simulated results meanings are

well articulated by Ackermann et al. (2002). Fcontr and Fconv have the same meanings as

1 in the nomenclature and figure 3-2, for the controlled vehicle and conventional

vehicle, respectively. Here s is the steering wheel angle.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.5

0

0.5

1

1.5

Time (Sec)

yaw

ra

te (

de

g/s

)

Steering wheel angle response

v = 50m/s, = 1

controlled

conventional

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Time (Sec)

Steering wheel angle response

s,

F [d

eg

]

Fcontr

s

Fconv

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.5

0

0.5

1

1.5

2

Time (Sec)

yaw

ra

te (

de

g/s

)

Yaw disturbance torque step response

v = 50m/s, = 1

controlledconventional

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.2

-0.15

-0.1

-0.05

0

0.05

Time (Sec)

Yaw disturbance torque step response

s,

F [d

eg

]

Fcontr

Fconv

s

Figure 4-2: Simulation results at 50 1v m s


44

4.3.2 Simulations of a conventional and a decoupled vehicle

All the simulations were based on the assumption that the vehicle was travelling on a dry

ground hence i was taken to be equal to 1. A value of 1500N for step lateral force was

always used while the disturbance yaw moment used was 1950Nm unless otherwise

stated.

For the conventional vehicle with front steering only, figure 4-3, the system shows that it

is oscillatory before coming to the steady state value for both the yaw torque and lateral

side wind disturbance input. As for the decoupled system, the responses show that the

system is able to arrest the continued rotation. The decoupled system with front wheels

being steered, figure 4-4, struggles a little bit as compared to the system that combines

with rear individual wheel steering, figure 4-5. Comparing graphs in figure 4-3 with

graphs in figure 4-4 and figure 4-5, one can notice that the controlled system removes the

effect of the disturbance, and this is evidenced by the zero steady state value, and also the

reduced peak value.

Figure 4-3: Step responses of conventional vehicle with front wheels steering only

0 0.5 1 1.5 2 2.5 30

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Time (sec)

yaw

ra

te (

rad

/s)

Yaw torque disturbance (M D ) input step response

0 0.5 1 1.5 2 2.5 30

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Time (sec)

yaw

ra

te (

rad

/s)

Disturbance lateral force (Fdy

) input step response


45

On comparing graphs 4-4 and 4-5, one will notice that graph 4-4 is oscillatory as

compared to graph 4-5 and their settling time is also slightly different. Of course it can be

noted that there was a price that graph 4-5 paid in that the peak is higher as compared to

graph 4-4. We should keep in mind that both of them are decoupled systems only that one

has both front and rear wheels steered while the other one has only the front wheels

steered.

Figure 4-4: Step responses of robustly decoupled vehicle with front wheel steering only

Figure 4-5: Step responses of robustly decoupled with front wheel steering and individual rear

wheel steering

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Time (sec)

yaw

ra

te (

rad

/s)


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.02

-0.01

0

0.01

0.02

0.03

0.04

Time (sec)

yaw

ra

te (

rad

/s)

Disturbance lateral force (F d y ) input step response

0 1 2 3 4 5 60

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Time (sec)

yaw

ra

te (

rad

/s)


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Time (sec)

yaw

ra

te (

rad

/s)

Disturbance lateral force (Fd y ) input step response


46

The control laws for the rear wheels that were used in all simulations from section 4.2.1

and section 4.2.2 were 3 3 6def

sK r

s

and 4 4 6def

sK r

s

with 3K and 4K as

desired damping gains and r as the yaw rate and the meanings are described in section

4.2.3. 4 is for the rear right and 3 is for the rear left. The constant used were not the

same and the value used for 3K was 39.6723 10 and the value for 4K was

38.9593 10 . This made it easy to feed signals to left and the right wheels because of its

simplicity. This control law was chosen and used because of its simplicity. Ackerman et

al (1992, 2002) suggested and used something similar to this control law.

The rear wheel steering control angles that are proposed here cannot be accommodated

within the theory derived, but the more general equations that can be used are given in

section 4.3.3.1.

All the cases that have been presented so far under this section assume that all parameters

remain constant and that only the disturbance input to the system changes i.e. either the

input is yaw torque, DM , or lateral force, dyF . Figures 4-6 to 4-9 show the yaw rate

response of the vehicle when subjected to a lateral disturbance force under the current

control laws (unilateral decoupled vehicle response with front wheel steering only), but

with varying friction coefficients, the same friction coefficient being used at each wheel.


47

Figure 4-6: Force input response 1 Figure 4-7: Force input response at .0 9

Figure 4-8: Force input response at .0 7 Figure 4-9: Force input response at .0 4

From the figures 4-6 to 4-9, it can be seen that as is reduced, the maximum yaw rate

increases and similarly, as is increased, maximum yaw rate amplitude also reduces. As

becomes smaller, the system becomes oscillatory and the converse is also true.

4.3.3 New control laws

There was need to come up with new control laws to enhance the concept of individual

wheel steering so that one can control the wheels independently of the other. The whole

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Disturbance lateral force (F d y) input step response

Time (sec)

yaw

ra

te (

rad

/s)

= 1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

yaw

ra

te (

rad

/s)

Disturbance lateral force (F dy) input step response

Time (sec)

= 0.9

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1 Disturbance lateral force (Fdy ) input step response

Time (sec)

yaw

ra

te (

rad

/s)

= 0.7

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Disturbance lateral force (Fd y ) input step response

yaw

ra

te (

rad

/s)

Time (sec)

= 0.4


48

idea is to have a method of controlling the wheels that could be steered differently from

each other.

4.3.3.1 Requirements for Control Law Design

According to the understanding of this author, a control law is merely a mathematical

statement that helps in decision making within a control system. It not only "helps", but it

also enforces (or at least tries to enforce) a certain dynamic behaviour. Consider a single

steerable wheel. Let’s say under certain conditions, a driver needs to generate a high

lateral force from the wheel. Ordinarily he will have to increase the steering angle in

order to generate the required steering force. The following were the control laws that

were used in the simulations under this section.

11 12 5 10c

ss

s s

[4.13]

22 12 5 10c

ss

s s

[4.14]

33

23 3 13 6defr s

k k s

[4.15]

44

24 4 14 6defr s

k k s

[4.16]

with

1DP

def

l as r sr s

v

[4.17]

def refr r r [4.18]

The chosen control laws for the rear wheels will ensure that the steering angle increases

when there is an increase in the coefficient of friction and reduces the steering angle

when the coefficient of friction is reduced. This is true provided that 23k and 24k are large


49

relative to 3 13k and 4 14k , respectively. As for the two front steering angle control laws

(equations [4.13] and [4.14]), the choice of the control laws was done to ensure that these

angles return to their zero position in the steady state. These control laws are of the form

of the fading integrator. According to Ackerman et al. (1992, 2002), the use of the fading

integrator is to make sure that the corrective steering returns to zero in the steady state

after the occurrence of the disturbance input. The whole idea is to only have this

occurring for the first second after disturbance input before the driver reacts. It is further

stated that this is desirable and done in order to achieve the same stationary cornering

behaviour that the driver is used to and also to unload the actuator for the corrective

steering. The other notable thing with the usage of fading integrators according to

Ackerman et al. (2002) is that stability problems in the yaw motion are reduced. There

are two types of the fading integrators used by Ackerman et al. (2002) which are second

order and first linear order filters.

In view of the new control laws, equation [3.37] was reworked. Using 1 1s c and

2 2s c equation [3.37] now becomes:

11 1 2 2 3 3 4 4

1 2 2

gp1 1 2 2 3 3 4 43 4 3

4

10

C

1Cy

s c

ds c

r r D

z z z z z z

C C C CFC mvmv mv mv mv

la C a C l C l CC rr MI I I I I I

11 1 2 2 3 3 4 41

1 2 2

gp1 1 2 2 3 3 4 43 4 32

4

10

C

1C y

cs

cd

r r

Dz z z z z z

C C C Cx

C mvmv mv mv mv Fla C a C l C l CC rr x MI I I I I I

Where:


50

1 1 2 21

C Cx

mv

1 1 2 2

2z

a C Cx

I

In all the simulation cases, the value used for step disturbance side (cross) wind force is

1500N and the fading integrator used is 2 5 10

s

s s .

4.3.4 Simulations of different coefficient of friction

Figures 4-10 and 4-11 are vehicle responses when the all the four wheels have the same

coefficient of friction . All the wheels are assumed to be under the same road

conditions.

4-10 Force input response 1

4-11 Force input response at .0 3

Comparing figures 4-10 and 4-11 to figures 4-6 to 4-9, it can be seen that the oscillations

have reduced when considering lower coefficients of friction.

The results given in figure 4-12 show rear wheels steer angle responses. The simulations

that were done in this case are where the left hand side of the vehicle is on the different

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.005

0.01

0.015

0.02

0.025

Time (Sec)

Yaw rate response to disturbance force (F d y ) input

yaw

ra

te (

rad

/s)

= 1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

Yaw rate response to disturbance force (Fdy ) input

Time (sec)

yaw

rat

e (r

ad/s

)

= 0.3


51

road conditions compared the right hand side. For example, the left hand side wheels

could be in mud while the right hand side wheels could be on dry tar mark road. We are

going to monitor what will be happening to the rear wheels ( 3 and 4 ). The coefficient

of friction for the left wheels, corresponding to 1 and 3 , will be the same and will

be varied; at the same time, the coefficient of friction for the right hand side wheels,

corresponding to 2 and 4 , will be the same and it will remain constant at 1.

Figure 4-12: Vehicle disturbances response to different

The results as seen in figure 4-12 show that when two sides are on different road

conditions, the model is able to turn more on the hard side (larger ) and it is also able to

turn less on the side with lower value of coefficient of friction. This shows that it is

possible to control each individual wheel independently.

0.5 1 1.5 2 2.5 3-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035 Rear wheels angles response to disturbance force (F dy

Time (sec)

Rea

r w

heel

ang

les

3 a

nd 4

[ra

d]

v = 50 m/s, 1

, 3 = 0.8; 2

, 4 = 1

3 4

) input

0 0.5 1 1.5 2 2.5 3-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Rear wheels angles response to disturbance force (F d y ) input

Time (sec)

Rea

r w

heel

ang

les

3

4 [

rad]

v = 50 m/s,

1 , 3 = 0.5; 2 , 4

= 1

3 4

and

Chapter 5 Implementation of Individual Wheel Steering

52

5 IMPLEMENTATION OF INDIVIDUAL WHEEL STEERING

5.1 Introduction

Theoretical modelling and analysis of an individual wheel steering (IWS) system was

done when all information was gathered both from available literature as well as from the

existing 4WS experimental vehicle that is in the SASOL Lab. This vehicle, figure 5-1,

used to have the normal mechanical front wheel steering while the rear wheels were

controlled through steer by wire. The whole 4WS was reworked in order to achieve

individual wheel angle steering. Electric actuators were installed for both front and rear,

on each wheel, to replace the previous steering system that included electrically actuated

hydraulic power steering on the two rear wheels only.

The rear wheel steering rack with power steering was disabled. Both the front and the

rear tie-rods were replaced by linear actuators. These were designed and built specifically

for this project and the purpose of these linear actuators were to effect the individual

wheel steering control. The normal front wheel steering via the rack and pinion system

was still partly available for the driver to control the front wheel steer angles.


53

Figure 5-1: Experimental vehicle

5.2 Motor characterisation

A Bosch wiper motor, including its reduction worm gear box, as used in a BMW 325i

model, as shown in figure 5-2, was used in each of the linear actuators. Furthermore, it is

worth mentioning that one can use either of two methods to control a DC wiper motor: an

armature control or pulse width modulation (PWM). Inside the position feedback control

loop, in the forward path, the DC motor and its control works as a speed control sub-

system, regardless of whether one uses armature control or PWM. The DC motor in

speed control can mathematically be modelled as an armature controlled DC motor and

can be represented as a first order equation as shown in equation [5.1], after


54

approximating a aL R , as given by Dorf et al. (2001), equation [2.69], where K and

can be determined experimentally.

1a

s K

V s s

[5.1]

Figure 5-2: Wiper motor sketch

Because of lack of information about this wiper motor, several tests were conducted to

determine its characteristics, primarily the gain K and time constant as shown in

equation [5.1]. Figure 5.3 shows the setup for the determination of the motor

characteristics. A tacho-generator was connected to the shaft of the motor, to generate a

voltage output signal proportional to the motor speed. The output from the tacho-

generator and the input from the motor were measured using a digital oscilloscope. The

data were extracted from the oscilloscope using a computer. Tacho-generator was used

only for the purpose of the characterization exercise. It does not form part of the actual

actuator control system.


55

Figure 5-3: Motor tests setup

Figure 5-4 shows the response of the motor to step input. The first experiments done were

to determine motor time constant .This was done in order to find out if the wiper motor

to be used meets the desired requirements.

-0.5 0 0.5 1 1.5 2-8

-6

-4

-2

0

2

4

6Response to step input

Time (s)

Am

plitu

de (

volts

)

Source (step) input

Response

Figure 5-4: Motor response to a step input


56

Figures 5-4 shows motor response to a step input and amplitude is in volts. The response

graph has been multiplied by the reciprocal of the DC gain, which is why in the steady

state the two graphs fall on top of one another. From this figure, it can be noted that the

motor attained its steady state in less that 1 sec. This exercise was done to make sure that

motors used fell within the specifications one of which was to be able to attain its steady

state in less than 1 sec. This was open loop input – output relationship. To find the

settling time of the system, locate the time on the plot when the magnitude crosses the

desired percentage of the final value. For instance to find the 10% settling time of this

system, look for where the response reaches 90% of the final value. The time constant,

, of the system is the time at which the response is 1 1/ e times by the final value.

The relationship between the time constant and the pole of a system is: pole = -1 / .

Other handy approximate relations for finding the time constant are:

= 10% Settling Time / 2.3 or

= 5% Settling Time / 3 or

= 2% Settling Time / 4.

From the graphs, the average was found to be equal to 0.1339 sec. This was deemed

acceptable.

The next step was to find the DC gain, K , of the motor. The DC gain is the ratio of the

steady state step response to the magnitude of a step input. The steady state step response

can be determined from the plot of the step response like one shown in figure 5-4. The

value of gain K was found after taking into consideration the calibration value of the


57

tacho-generator as well as the gain from gear ratio of the wiper motor gear box. The

average value of DC gain that was found was 9.972 / /rad s v .

5.3 Potentiometer characterisation

For the linear actuator to work properly there was need to create a position control

system, with position feedback of some sort. After considering all the possibilities, it was

decided to measure the angle of rotation of the output shaft of the wipe motor gearbox for

the purposes of position feedback. A multi-turn potentiometer was used for that purpose

as an angular position feedback transducer. Figure 5-5 shows the assembly used in

finding the characteristics of the multi-turn potentiometer that was used as an angle

feedback sensor. The body of the Potentiometer is fixed to the wiper motor chassis to

prevent it from moving. The shaft of the potentiometer was inserted into a cork coupling

that was fixed to the base as shown. When the motor is turning, it rotates the shaft of the

Potentiometer. As the shaft turns, the voltage output can be measured and converted to

the angle that the sprocket has turned.


58

Figure 5-5: Potentiometer connection sketch

The unit shown on figure 5-6 shows the actuator as assembled to the vehicle.

Figure 5-6: Actuator unit


59

Experiments were done and the relationships below were found by measuring

potentiometer angle, wheel angle and voltage output from the potentiometer. The

calibration of the potentiometer was done by using the dividing head and indexing plate

in the machine shop. The base of the potentiometer was clamped while turning the

indexing plate. The potentiometer had 10 complete turns. Voltage measurements were

taken for every complete turn of the knob of the potentiometer. The results were plotted

for voltages against potentiometer angle.

The next stage was to measure the wheel angle and then measure the potentiometer

voltage. This was done to find the relationship between the potentiometer voltage and the

wheel angle. The first step was to zero all the wheels by aligning them. When the wheels

are aligned, then that is the zero point for the potentiometer. A hard paper was stuck to

the ground and then a straight edge was used to mark a straight line on the paper as the

zero angle. The wheel was then turned using the actuator. The turning was not done

continuously; it was done by starting and stopping the actuator. Every time when the

actuator was stopped, successive lines were drawn using a straight edge on the paper that

was stuck to the ground. Voltage measurements from the potentiometer were taken at

every stop. This was done in both directions of the wheel turning and the same procedure

was repeated on all the four potentiometers and all the four wheels. Angles were read off

from the paper later on. The measured potentiometer voltage values for the wheels angle

were plotted.

A mathematical relationship stemming from linear progression between the turn of the

potentiometer knob for voltage and wheel angle was developed, and this was included in


60

the main algorithm. By using Matlab, the equation that came out was 0.28 0.017y x

where y is the wheel angle and x is the potentiometer voltage.

The graphs in figures 5-7 and 5-8 show the relationship between potentiometer voltage

and Potentiometer angle as well as the relationship between potentiometer voltage and

wheel angle.

Figure 5-7: Graph of potentiometer angle against potentiometer volts

0 500 1000 1500 2000 2500 3000 3500 40000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

pot (degrees)

Pot

entio

met

er v

olts

Front left wheel potentiometer volts against potentiometer angle


61

Figure 5-8: Graph of wheel angle against Potentiometer volts

5.4 DC motor controller

The circuit in figures 5-10 and 5-11 were built to provide an interface between the

computer and the motor. Power is supplied from the car battery through point a (in the

red circle). The signal that controls the speed of the motor enters through point b. Part A

(in green) generates the pulses for the PWM and passes these to the LMD18200 chip,

which drives and controls the speed of the motor. The PWM was used for the DC motor

speed control sub-system. The circuit to generate pulses uses two 555 ICs connected as

shown. A 7805 regulator is included to restrict voltage to and from part A to 5 volts

because the computer voltage signals can only go to the maximum of 5 volts. Pulses from

part A enters part B (in green) through point d. The decision from the computer to change

direction enters the chip at pin number 3 through point c. This tells the chip to turn the

motor either clockwise or anticlockwise.

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3 Front left wheel steering angle against potentiometer volts

Wheel angle (degrees)

volts


62

DC Motor Controller

DC Motor Gain

GainSampler

A/DPID

ControllerSampler

D/A

Referencesignal

Potentiometer

Sprocket angle

Computer

iGain

Lead screw

speed

directionWheel angle

Figure 5-9: Actuator block diagram.

Figure 5-9 shows a single actuator block diagram. All that is inside the dashed line was

implemented in the computer and a single computer was used for all four actuators and

also to run the control algorithm, which determined the input/reference signal to all four

actuators. The gain after the integrator is the gear ratio of the worm gear. Downstream of

the point where the sprocket feedback angle is picked, there is a gain block containing the

pitch of the lead screw that translates the sprocket rotation angle into a rectilinear

displacement, (the chain drive had a gain of 1).


63

31TIP C

Transistor

220k

1k

1k

470k220 F

1k

1 414N

DIODE

0V

12V

8

51

74

1 5

8

2

3

47

2,63 6

1k

0.1 F0.1 F 0.01 F

M12 V DC Motor

ReverseForward

31

1110

5

2220 F

1 F6

7LMD18200

From computerFor direction

4

From computerfor proportional

Voltage

7805

10k

1 F

1 F

a

b

c

e

d

555555

Regulator

PWM

Figure 5-10: DC Motor Controller circuit diagram

Figure 5-11: DC motor Controller circuit box


64

A cooling fan was used to cool down the LMD18200 chip, which otherwise overheats

and then shuts itself down

Figure 5-12: Illustrating the principle of Pulse Width Modulation, National Semiconductors

(2005)

Figure 5-12 shows the concept of PWM. If the motor is to turn at full speed, the system

has to supply 100% duty cycle pulses. In order to reduce the speed of the motor, there

should be a supply of pulses as shown in figure 5-12 depending on the desired speed.

LMD

1820

0

Figure 5-13: LMD18200 chip, National Semiconductors (2005)

Figure 5-13 shows an LMD18200 chip which works as an H-Bridge as illustrated in

figure 5-14.


65

Figure 5-14: LMD18200 circuit diagram, National Semiconductors (2005)

5.5 Actuator assembly

The whole actuator comprised the DC motor controller as well as the actuator assembly

as shown in figure 5.15 (see also figure 5-6). The actuator assembly consisted of the

Bosch DC motor, a chain, an internal threaded hub and housing, a lead screw, and a

bearing as the major components, as shown in the exploded view sketch of figure 5-16.

On the inboard side, the actuator was connected to the chassis, in the case of the rear

actuators, or the steering rack, in the case of the front actuators, via a universal joint,

while the outboard end was connected to the hub assembly via a spherical joint, similarly

to a normal tie rod end connection. The inboard universal joint was used, rather than a

spherical joint, to support the actuator against possible rotation about its axis, for example

due to the offset weight and reaction torque of the DC motor. The lead screw and the

internal threaded hub were arranged in such a way that the lead screw could move in and

out of the actuator assembly as the motor is rotating. As can been seen from figure 5-16,

the actuator is not sealed. This was due to space limitations, and because of the

preliminary and experimental nature of the setup.


66

Figure 5-15: Actuator – hub assembly connection

Figure 5-15 shows one actuator, connected to the right rear wheel, replacing the right rear

tie rod of the original four wheel steering system.


67

Figure 5-16: Actuator exploded view

Chapter 6 Experiments and Results

68

6 EXPERIMENTAL SETUP

This chapter describes the experimental setup, experimental tests that were carried out

and the results that were obtained. Experimental tests were done for the individual wheel

steering system. These tests were before expanding the mathematical model and

therefore the tests did not implement the new control laws that are described in equations

[4.13] to [4.18]. In other words, control laws equations [4.13] to [4.18] were never tested

experimentally. More also, the experimental setup, as depicted in figure 6-1, used control

law equation [3.50] for the front wheels and the rear wheel control laws mentioned on

page 47, below figure 4-9.

r

yDPa

2c

3

4

DM dyF

s refr

1c

Figure 6-1: A sketch of experimental setup


69

Figure 6-1 shows the experimental setup diagrammatically. Each wheel was controlled

individually via the computer. The feedback signal from lateral acceleration shown was

not used for any purpose although it is one of the outputs as shown in equation [3.70]. A

yaw rate sensor was connected on the vehicle to measure the yaw rate for feedback

purposes but also to have a measurable result.

Figure 6-2: Computer

Figure 6-2 shows a computer that was used for experiments. The output of this computer

can be both analogue and digital, because it was equipped with an analogue to digital

card. In this experiment, one analogue and one digital output was used for each steering

actuator. The analogue output was used for control voltage while the digital output was

used to switch direction of the dc motor in the actuator. The output voltage from the

computer was used to change the speed of the motor. If the digital output was 0 the motor

turned in one direction and if the output was 1 the motor turned in the opposite direction.

Computer


70

6.1 Experimental Results

There were two sets of experiments that were conducted to validate the proposed theories

as will be discussed in section 6.1.1 and section 6.1.2. The driver kept the steering wheel

straight in most of the tests that were done. The exceptions were on a number of tests

where the driver had to induce sinusoidal input from the steering wheel. This was to

ensure that there is separation of the driver task and that of control system. The driver’s

job had to do with path tracking while the control system had to control the yaw rate.

6.1.1 No input from the steering wheel

It was decided to employ uneven braking between the left and right wheels to induce a

disturbance moment on the vehicle during road tests. For this purpose the brake calliper

of the left front wheel was taken off the disk, thus disabling braking on that wheel. The

driver then had to brake hard to induce a disturbing moment. The advantage of using this

method to generate a yaw moment is that it is cheap. On the other hand, it is unlikely that

this scenario will be encountered in a real life situation. It was done purely for

experimental purposes. Instead, it would have been better to use wind generators to create

side wind disturbance, but such generators were not available. The results provided in

figures 6-3 to 6-9 are from the test run that was seen to be successful. The only known

parameter is the speed since the successful test was done at a speed of 50 km/h at the time

of braking. It was difficult to measure other parameters like the generated yaw torque

from braking. The other tests were partially successful while other tests were not very

successful as will be highlighted later.


71

Figure 6-3: Controlled vehicle yaw rate response.

Figure 6-3 shows the comparison between simulated and experimental results. The brakes

were applied when the vehicle was travelling at a speed of 50 km/h. Because of lack of

other information, and lack of measuring methods, some of the parameters were assumed

for simulation purposes. For example, to know how much brake force and the size of

yaw moment created during the period that the brakes were applied, were difficult.

Figures 6-4 to 6-7 show the comparison between the experimental and the simulated

results of each wheel.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -5

-4

-3

-2

-1

0

1

2

3

4

5

Yaw

Rat

e (o

/sec

)

Time

Yaw Rate Response With Control

Experimental Results

Simulation Results


72

Figure 6-4: Front right additional steering angle response

Figure 6-5: Front left additional steering angle response

1 1.5 2 2.5 3 3.5 4 4.5 5 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

Time

Ste

erin

g A

ngle

s (

Deg

rees

)

Steering Angles Results

c1 ---> Experiment

c1 ---> Simulation

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4 Steering angles results

time

Ste

erin

g an

gle

s (D

egre

es)

--> Experiment

--> Simulation


73

Figure 6-6: Rear right steering angle response

Figure 6-7: Rear left steering angle response

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

Time

Ste

erin

g A

ngle

s (

Deg

rees

)


4 ---> Experiment

4 ---> Simulation

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

Ste

erin

g A

ngle

s (

Deg

rees

)

Time


3 ---> Experiment

3 ---> Simulation


74

6.1.2 Sinusoidal input from the steering wheel

Figures 6-8 and 6-9 show the yaw rate response of the test vehicle to sinusoidal input that

was done through the steering wheel with the control system either inoperative (figure 6-

8) or operative (figure 6.9) and the vehicle travelling at 50 km/h. The reason why a

sinusoidal input test was done was because it was thought that the controlled system

could be fooled to think that the yaw that resulted from the driver giving a sinusoidal

steering input was actually some disturbance that it should reject. Hence we expected

that a successful test would be indicated by a reduced amplitude in the sinusoidal yaw

rate variation, in the case of the controlled car. That the yaw rate would reduce to zero

after the test is not an indication of success, because any stable motorcar will have a zero

yaw rate in the steady state after a sinusoidal steering input was removed. From figure 6-

9, it can be seen that the controlled car has a smaller variation in yaw rate than the

uncontrolled car, figure 6-8. The test driver tried to use the same magnitude of sinusoidal

excitation in the two cases, even though this is a subjective measure.


75

Figure 6-8: Uncontrolled vehicle test result at 50 km/h

Figure 6-9: Controlled vehicle test result at 50 km/h

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-8

-6

-4

-2

0

2

4

6

8

10

12Yaw rate sinusoidal input response - Uncontrolled

Time (sec)

Yaw

rat

e (D

egre

es/s

)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1.5

-1

-0.5

0

0.5

1

1.5

2Yaw rate sinusoidal input response - Controlled

Time (sec)

Yaw

rat

e (D

egre

es/s

)


76

There were other tests that were done at lower speeds, the results of which showed that

the rear wheels did not respond satisfactorily, as shown in the figure 6-10. It was

observed that at lower speeds, the rear wheels failed to respond presumably because of

weight. The engine of the vehicle is mounted at the rear and as a result it was hard for the

wiper motors at the back to steer the rear wheels at the lower speeds. It can be noted that

the response from the front wheels is very small. This may also mean that the command

signal to the two rear actuators was within the dead band of the rear actuators. The

higher vertical wheel loading on the rear wheels due to the engine weight would have

caused a larger dead band in the case of the rear wheels, compared to the front.

Figure 6-10: Steering angles response at low speed (20 km/h)

Figure 6-10 shows a test result that was conducted at an approximate sinusoidal

disturbance input through the steering wheel while the vehicle was travelling at a speed

of 20 km/h. The 3 and 4 curves are not responding the same perhaps because the

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Time

Ste

erin

g A

ngle

s (

Deg

rees

)

Steering Angles - Experimental Results

c 1 c 2 3

4


77

vertical wheel loading are not the same (see the measured loads shown on the very last

page of this report) bearing in mind the low speed at which the vehicle was travelling.

Figure 6-11: Steering angles responses to braking at 40 km/h

Figure 6-11 shows the steering angles recorded while disturbing the vehicle with braking

from 40 km/h. It shows that all the wheels managed to return to zero steer position

though quite slowly.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6 Steering Angles - Experimental Results

Time

Ste

erin

g A

ngle

s (

Deg

rees

)

c1

c2

3

4

Chapter 7 Conclusions

78

7 CONCLUSIONS AND FUTURE WORK

The research work has measured and presented some results, of which the conditions to

make a full correlation study between measurement and numerical prediction is not

known, but at least the work can show simulation results that quantitatively look quite

similar, and the conditions that pertain to this simulation is quite realistic, i.e., it would

have been possible to have made an experiment with matching conditions.

It can be noted from the theories presented that under dissimilar conditions at the wheels,

in order to give a true picture of the actual vehicle, every wheel has to be treated

individually. When one is driving or steering, wheels are in contact with the ground

independently. Even the load (weight) distribution of the vehicle is not the same on all

the wheels as evidenced from the measurement results of the experimental vehicle. When

wheels are not treated as a single entity, like in the bicycle model, some information is

bound to be lost. In this dissertation a steering control analysis has been presented,

similar to that of Ackermann, but with each wheel treated on its own. In addition to this,

the modelling of a side wind acting at the centre of pressure was included in the analysis.

The model that has been investigated is clearly an improvement of the bicycle model that

has been used extensively by researchers in the understanding the concepts of 4WS. The

results from simulations indicate that now it is possible to control each wheel

independently. We have seen an improvement in the vehicle response by individually

controlling the wheels as compared to the conventional steering. With control laws based

on the bicycle model, the vehicle tends to oscillate, especially at lower coefficients of

friction. The independent steering control laws in this work were shown to result in larger

stability margins (fewer oscillations) even at lower coefficients of friction. The cases that


79

were investigated in this work are generic in that they were not condition specific,

because it is now possible to investigate different road conditions at the left and right

hand side wheels.

From the results presented, the wheels are able to return to their zero steering angle

position after the effect of a disturbance was rejected. It was important to make sure that

no steering angle remains after action of the control system. This highlights the ability of

this experimental vehicle to reject disturbances.

There were some problems with the experimental results obtained. The results were

deficient in terms of repeatability because of the configurations and characteristics of the

wiper motors used. Firstly, there were slight differences between clockwise and anti-

clockwise characteristic responses of the wiper motors used. Secondly, at low speeds, the

motors could not push hard enough to generate the required angles. Another problem was

that the yaw rate sensor used had significant inherent noise. Otherwise, given all

conditions to be ideal, with this method, there is a possibility to achieve disturbance

rejection.

Since one of the objectives was to be able to steer the wheels individually, and this was

met, one can take advantage of that to further the research. On the experimental side, it

will now be easier to let the wheels be on different road conditions and monitor the

performance. More also, one can do further tests like subjecting the vehicle to different

road manoeuvres e.g. double lane changes, parallel parking, and side gust response

measurements, if side wind generating tunnels were to be made available. Future work

can benefit from employing more suitable actuators, because the type of motors that were


80

used were essentially underpowered and the performance and response of the motors in

clockwise and anticlockwise directions were observed to be different. One can also think

of improving the system modelling by using the equations in their nonlinear form,

especially with respect to simulation, and see if there will be any improvements (or

perhaps deteriorations) compared to the linearized equations analysis. The extension of

this work into nonlinear regime should also look at more sophisticated tyre models. The

vehicle dynamics modelling also needs to be expanded to include roll dynamics, with the

associated load transfer. This will be important if manoeuvres like double lane changes

were to be simulated. More also, the control laws [4.13] to [4.18] that were investigated

in simulations, assumed that the friction coefficient at each wheel is known. It is,

however, quite difficult (if not impossible) to measure, in real time, the friction

coefficient in order to implement these control laws. This is one of the major

disadvantages of the proposed control laws. Therefore, one can also do a research and

look into practical ways on how to estimate the friction coefficient, if the uses of these

control laws are to be further employed.

References

81

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Abe, M., and Mokhiamar, O. (2002). Active wheel steering and yaw moment control

combination to maximize stability as well as vehicle responsiveness during quick lane

change for active vehicle handling safety. Proc Instn Mech Engrs, Vol. 216 part D: J

Automobile Engineering, pp. 115-124.

Ackermann, J. and Bunte, T. (1997). Yaw disturbance attenuation by robust decoupling

of car steering. Control Engineering Practice, (pp.1131-1136).

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2590.

Ackermann, J., Bunte, T., and Odenthal, D. (1999). Advantages of active steering for

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Ackermann, J. and Bunte, T. (1996). Driving safety by robust steering control. Proc. Int.

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Ackermann, J., in co-operation with Blue, P., Bunte, T., Guvenc, L. Kaesbauer, D.,

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Hucho, W.H. (1987). Aerodynamics of road vehicles. Butterworth. Printed in Great

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Appendix

85

APPENDIX

Appendix B Vehicle Information

86

Table B-1 shows experimental vehicle information. Some of the values here were from

direct measurement of the vehicle while some of the values were calculated using

available information that was gathered by Klein (1996).

Table B-1: Vehicle data

Variable Description Value Unit

1C Front left cornering stiffness 12682 N / rad

2C Front right cornering stiffness 11414 N / rad

3C Rear left cornering stiffness 19023 N / rad

4C Rear right cornering stiffness 20502 N / rad

m Total mass 737 kg

fl Front axle distance to C.P. 0.678 m

gpl distance from C.G to C.P. 0.772 m

a Front axle distance to C.G. 1.3 m

rl Rear axle distance to C.G. 1 m

l Wheel base 2.3 m

t Half track width 0.72 m

zI Yaw inertial 1320 2kgm

cgh C.G. height 0.57 m

Appendix B Vehicle Information

87

The values below are the static vertical wheel forces of the experimental vehicle that

were found after measuring with wheel scales:

1 146.03zF kgf

2 139.68zF kgf

3 210.431zF kgf

4 244.44zF kgf

use of individual wheel steering to improve vehicle

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